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The relationship between the maximum in the rate of
pressure rise in the aorta and left ventricle
Jesse Mesman, 5744318
19 Augustus ,2011
Supervised by Dr. Ir. Maria Siebes and
MSc M.C. Rolandi
Academic medical centre,
Biomedical engineering and Physics,
University of Amsterdam
Report of the Bachelor's Project Physics and Astronomy
Accreditation: 12 ECTS
Performed in the period 25-05-2010 - 25-07-2010
Second Reader: Ed van Bavel
Contents
1
Scientific Summary
3
2
Introduction
4
3
Method
7
4
Results
12
5
Discussion
25
6
Conclusion
26
7
References
26
8
Appendix A
27
9
Appendix B
28
2
1
Scientific Summary
The heart is the most vital organ of every human being. It is responsible for circulating blood
through the body, supplying oxygen to the organs and removing carbon dioxide from the
organs. Because of this important function of the heart we would like to measure the
contractility of the heart. The contractility of the heart is a measure for the power with
which it contracts. To check the contractility the usual method is to measure the maximum
of the rate of pressure rise in the left ventricle.
When the heart pumps blood into the aorta the pressure in the aorta rises depending on the
ventricular and vascular properties. If the contractility of the left ventricle is increased then
the rate of aortic pressure rise also increases. Arterial properties also factor in how the
pressure in de aorta develops. Because of this it is likely that there is a relationship between
the rate of pressure rise in the aorta and the left ventricle depending on vascular properties.
If we could find a relationship between the maximum rate of pressure rise in the aorta
(dPa/dtmax) and the maximum rate of pressure rise in the left ventricle (dPlv/dtmax) we could
approximate the contractility by measuring the pressure in the aorta. This would help
because the pressure in the aorta is normally always measured when a patient is diagnosed
in the catheterization laboratory.
This research tried to find the relationship between the maximum in pressure rates by using
data from 7 patients which have already been obtained before this paper. The pressure
measurements in this dataset were taken when the patients were in rest, while the patients
were performing the Valsalva maneuver and when the coronary arteries of the patients had
increased blood flow. A separate measurement determined the same data but with 2
patients in rest and had their heart rate artificially heightened by fifty percent. From these
datasets the rate of pressure rise, the aortic pulse pressure and heart rate were calculated
with a program I created.
In all conditions dPa/dtmax and dPlv/dtmax are found to be affected by the pulse pressure in
the aorta and the left ventricle. In the hyperemia and baseline conditions these properties
are not influenced by the heart rate. When the patient is performing the Valsalva maneuver
these properties are affected by the heart rate because the heart rate is dependent on the
pulse pressure.
With these dependencies taken into account statistic analysis was done to relate dPa/dtmax
and dPlv/dtmax to the heart rate and/or aortic pulse pressure. In all conditions the same
conclusion was found: dPa/dtmax does not relate significantly enough to dPlv/dtmax through
the aortic pulse pressure and/or the heart rate to be used in clinical application. This
relationship cannot be used to determine the contractility of the heart accurately enough
with only the pulse pressure and heart rate.
3
2
Introduction
For a human in rest the heart which is made of cardiac muscle contracts at a steady rate to
pump blood in the aorta. Because of the force of contraction a pressure is exerted on the
walls of the blood vessels. To understand how there could be a relationship between
dPa/dtmax and dPlv/dtmax this section first describes the flow of pressure during the cardiac
cycle in these blood vessels. The cardiac cycle describes the events related to the contraction
of the heart during one beat. Next the properties influencing these pressures are described
and related to the relationship we would like to find. Also the influences of the adenosine
and Valsalva maneuver on the pressure in the blood vessels are explained. In the last section
a short literary review is given.
2.1
The cardiac cycle
First of in figure 2.1 below the heart is shown with all its components.
Figure 2.1 The heart with all its components and connections to the arterial and venous circulation.
The left and right halves of the heart contract simultaneously so the events of the cardiac
cycle are the same for both halves. In this paper the focus is on the left part of the heart
because of the connection to the aorta. The cardiac cycle consists of two major stages,
diastole and systole.
The cardiac cycle begins with atrial systole. In atrial systole the atrioventricular valve is open
and the muscle in the left atrium contracts thus pushing blood into the left ventricle. This
causes the pressure in the left ventricle to rise slightly. Figure 2.2 below shows the pressure
in the aorta and in the left ventricle over a timeframe of several heartbeats. This is
accompanied with an ECG where the first peak in the ECG signals the beginning of atrial
systole. The pressure graph illustrates the slight rise in pressure in the left ventricle.
The second phase of the cardiac cycle is ventricular systole. The atrioventricular and
semilunar valve is closed and the ventricular muscle contracts isovolumetrically. The rise of
pressure in the ventricles causes the semilunar valve to open and the blood flows into the
4
aorta. In this steep rise of pressure the maximum in rate of pressure rise occurs in the aorta
and the left ventricle. This sequence of pressure differences can be seen in figure 2.2 where
the second peak in the ECG signals the start of ventricular systole. In the pressure graph at
the start of ventricular systole only the ventricular pressure rises. When the semilunar valve
opens the aortic pressure also rises. When the blood is pumped out of the heart at the end
of systole the pressure decreases again and the semilunar valves close. Then in diastole
when the pressure in the ventricles drops lower than the pressure in the atria the
atrioventricular valve opens and blood seeps back into the ventricles.
Figure 2.2 ECG of a patient in the upper graph and in the lower graph the pressure in the aorta and in the left
ventricle.
2.2
Properties of blood vessels
Contractility is a measure of the fitness of the heart but what is contractility exactly?
Changing the contractility of the heart changes the force generated by the heart
independent of the preload. Preload is the length initial stretching of the heart muscle
before contracting. Aside from this, changes in contractility also alter the maximal velocity of
muscle fiber shortening at zero afterload. Afterload is the pressure the heart needs to
generate to pump the blood into the aorta. dPlv/dtmax Is proven to be a good approximation
for the contractility of the heart.1 This measurement is clearly dependent on the preload but
is still a reliable measurement whatever the shape or form of the blood vessels.
The pressure in the aorta and the rise in this pressure depend on the characteristic
impedance, the arterial compliance and the pulse pressure of the blood vessels and thus the
relationship we are interested in probably does too. So first let us look at a few
cardiovascular properties.
The pulse pressure (PP) of a blood vessel is the difference between the maximum and
minimum pressure during a cardiac cycle in this vessel. The total arterial resistance (RT) is
the total resistance of the arteries and is calculated by dividing the pressure difference over
the beginning and end of the systemic arteries by the cardiac output. Cardiac output (CO) is
the volume of blood per minute pumped out of the heart. Arterial compliance (C) is a
measure for the ability of the arteries to expand and contract with pressure changes. This
5
compliance can be calculated for blood vessels reliably by modeling the arterial system as a
two-element Windkessel consisting of a resistance and compliance in parallel. With this
model an expression can be found for the compliance depending on the pressure in the
arterial system and experimentally derived constants.2
The last important property is the characteristic impedance. Characteristic impedance of the
aorta is the input impedance of the aorta at a specific frequency. Characteristic impedance
can be seen as a link between the Windkessel model and the way the waves travel through
the arterial system. Characteristic impedance is equal to the wave speed multiplied by the
blood density divided by the area of the cross-section.
To find the relationship between the pressures it is best to include as many of these
properties as possible.
In the data used by this research the pressure measurements are taken when the patients
are in rest but also when coronary blood flow is elevated. The adenosine is injected into the
coronary artery causing increased blood flow (hyperemia).
For the final condition the patients performed the Valsalva maneuver. The Valsalva
maneuver is mostly used as test of cardiac function. It is performed by exhaling against a
closed airway causing pressure in the chest to rise. Because of the rise in pressure less blood
returns to the heart and thus the output of the heart is lower. When the pressure on the
chest is released the cardiac output increased rapidly till slightly above normal and then
stabilizes. The relationship between the maximum rate of pressure rise in the aorta and left
ventricle should hold in rest but also in these two conditions. In figure 3.2 below the
pressures in the left ventricle and aorta are shown during this Valsalva maneuver.
Figure 3.2 The pressure in the left ventricle (blue line) and aorta (red line) over time during the Valsalva
maneuver.
6
2.3
Literary review
In 1972 Taylor et al3 first looked at the relationship between dPa/dtmax and dPlv/dtmax and
found a close correlation between the two. But this relationship varied quite a lot over
differing conditions. Because they did not look at what caused this variance they did not
research this relationship any further.
A few others4-6 have tried to find a noninvasive determination of the dPlv/dtmax by using
Doppler signals of mitral insufficiency. They were quite successful but the proposed
relationships can only be used in patients with mitral regurgitation. The same approach was
taken in the research done by Rhodes et al7 but only uses the diastolic blood pressures and
isovolumic relaxation times and is thus independent of the patients disease.
A similar approach is taken in the research done by Masutani et al 7 where the relationship
between dPa/dtmax and dPlv/dtmax is found to have a significant and close correlation
through the Mean Arterial Pressure (depending on the aortic pulse pressure) and the
characteristic impedance. This relationship is investigated in healthy patients and in patients
with various diseases in various conditions like after atrial and dobutamine pacing. In all
conditions a close correlation is found. This research tried to find this same relationship but
with pulse pressure and heart rate as determinants of this relationship.
3
Method
To investigate the relationship between dPa/dtmax and dPlv/dtmax we need pressure data
from patients. First the method how these data are acquired is explained next how these
data are analyzed and finally the justification for the chosen derivative approximation.
3.1
Data acquisition
The pressure data was acquired in the catheterization laboratory prior to this research.
These pressure measurements were done on patients in healthy coronary arteries in three
conditions. First the measurements were done on patients rest. Second when adenosine was
injected into the blood causing increased blood flow. Finally the patients performed the
Valsalva maneuver. The measurements were done on seven patients in total.
In these conditions the following data were acquired: time t (s), pressure in the aorta
(mmHg), pressure in the left ventricle (mmHg), the voltage measured in the ECG (mV) and
various other measurements like heart rate (beats per minute) and maxima in pressure
(mmHg) taken over one beat. These data were taken at a sampling rate of 200
measurements per second and stored in text files for later analysis.
In a separate measurement the same data was obtained but with different conditions. In this
dataset the measurements were done with the patients being in rest and having their heart
rate artificially heightened by approximately 50 percent with a pacing catheter. These
measurements were done on 3 patients in total.
7
3.2
Approximating the derivative of pressure
Calculating the rate of pressure rise from the pressure data should be done as accurate as
possible. Approximating the first derivative of pressure out of the numerical pressure data
can be done with various methods. The easiest and most inaccurate method is by using the
difference. In this method the value of the derivative at point xi is the value of the function
at xi+1 subtracted by the value of the function at xi divided by the fixed interval h like in the
formula below.
In this research the Taylor series method is used to obtain the first derivative using a second,
third and fourth order polynomial. The Taylor series method is based on the fact that
functions can be expressed as an infinite series in terms of its derivatives and the fixed
interval between the data points. Then one can write the Taylor series around these data
points and solve for its derivatives with the values of the function.
One can derive the forward, central or backward derivative where the difference is the
points used from the function. In the forward derivative the derivative at xi is approximated
by using the values of the function at the later points xi+1, xi+2, … , xi+n with n depending on
the polynomial order. The central derivative uses the values of the function at points
previous and after the evaluated point and the backward derivative uses the values of the
points previous to the evaluated point.
To find the most accurate approximation for the derivative of our pressure curves one can
use a test function and apply the various sorts of approximation on this test function. One
knows the maximum derivative of this test function and can check which one of the
approximations is most accurate. This test function should resemble the pressure curves for
the best result. For example one can take as a test function:
In this function we know the maximum derivative to be 2.5 at x = 0. In this research the
derivative of the Fermi-Dirac function is approximated with the finite difference (1), a
second order polynomial (2), a third order polynomial (3) and a fourth order polynomial (4).
8
Below in figure 3.1 the results of these approximations for our test function are displayed.
Figure 3.1 Upper graph: Our test function. Lower graph: Approximation of the derivative of our test function
with finite difference (blue), a second order polynomial (green), a third order polynomial (black) and a fourth
order polynomial (red).
9
In the lower graph of figure 3.1 can be seen that the fourth order approximation has the
closest value to 2.5 of all the approximations. The values of the first and second order
approximations of the maximum derivative are both 2.375, the third order approximation
has a maximum derivative value of 2.637 and the fourth order approximation has a
maximum derivative value of 2.475. These maxima are all found at x = 0. The first forward
derivative using a fourth order polynomial utilized in this research is obtained by the method
shown in appendix A.
Below in figure 3.2 is displayed how the pressure in the aorta is approximated by finite
difference and a fourth order polynomial.
Figure 3.2 Upper graph: Pressure in the aorta (black) and left ventricle (grey). Lower graph: Approximation of
the derivative of pressure with finite difference (blue) and a fourth order polynomial (red).
3.3
Data analysis
The pressure data is analyzed with a self created program in Delphi. The code of this
program is in appendix B. In the program the rate of pressure rise in the aorta and left
ventricle is calculated with the previously stated approximation.
The values of the maxima in the rate of pressure rise in the aorta and left ventricle are
recorded together with their time of occurrence in the beat. Next the ratio between these
maxima in this same beat is calculated. The aortic and left ventricular pulse pressures are
calculated by subtracting the lowest pressure from the highest pressure during a cardiac
cycle. The heart rate in beats per minute is calculated per beat by dividing 60 by the length
of the beat in seconds.
Not all the vascular properties one would like to know could be calculated. The total arterial
resistance could not be calculated because the cardiac output was not measured. The
10
arterial compliance could not be calculated because the volume change in the arteries is
unknown. Finally the characteristic impedance could have been calculated by using various
simplifications and making assumptions about the body. But due to time constraints the
characteristic impedance is not included in this research.
To analyze the calculated pressure rates and vascular properties one looks for a relationship
between the ratio dPa/dtmax/dPlv/dtmax, heart rate and aortic pulse pressure. To model this
relationship multiple linear regression is used in SPSS. One uses the aortic pulse pressure as
a coefficient because in the introduction was seen that the aortic pressure and the rise in
this pressure depend on the aortic pulse pressure. This would imply that the ratio depends
on the aortic pulse pressure too. The heart rate is also used as a coefficient to determine if it
affects this ratio. The general equation used in multiple linear regression is:
In the case of this research the coefficients in this equation will give an indication how much
the aortic pulse pressure and the heart rate contribute to the ratio of dPa/dt max/dPlv/dtmax.
Out of the statistical errors and P-values of this model can be concluded if there is a
significant enough relationship between the rate of pressure rise in the aorta and left
ventricle to be used in clinical application.
11
4
Results
4.1
Influence of heart rate on dPa/dtmax
After compiling the data the first item looked at is the relationship between the heart rate
and the maximum in the rate of aortic pressure rise. Below in figure 5.1 this relationship is
displayed with the data used from the separate measurements done in patients with their
heart rate raised. As can be seen in the graph for the two patients BIV3 and BIV5 the heart
rate has no significant influence on the maximum in aortic pressure rise. It is thus reasonable
to exclude heart rate as a coefficient in the linear regression.
Figure 4.1 The maximum rate of pressure rise in the aorta plotted against heart rate for
two patients (BIV3,BIV5) at rest (squares) and with heightened heart rate (triangles). Data taken in baseline
(filled) and hyperemia (open).
4.2
Direct relationship between dPa/dtmax and dPlv/dtmax
Looking at the main data one can compare the calculated dPa/dtmax and dPlv/dtmax by the
used program and look at their direct relationship. In figure 4.2, figure 4.3 and figure 4.4
below these two maxima in rate of pressure rise are plotted against each other where the
measurements were done in the three conditions described in the method.
12
Figure 4.2 The measured dPla/dtmax plotted against the measured dPlv/dtmax for 7 patients in baseline.
Figure 4.3 The measured dPa/dtmax plotted against the measured dPlv/dtmax for 7 patients where the aortic
pressure is measured in a hyperemic artery.
13
Figure 4.4 The estimated dPlv/dtmax plotted against the measured dPlv/dtmax for 7 patients who are doing the
Valsalva Maneuver. These data points were measured when the patients were exerting a pressure on the
chest.
As can be seen in the figures above it is not possible to significantly relate dPlv/dt max directly
to the dPa/dtmax with a linear equation. The data points do not show any significant relation,
they look random. Therefore aortic pulse pressure will be used as a coefficient to determine
a significant relationship between dPa/dtmax and dPlv/dtmax.
4.3
Relationship between heart rate, pulse pressure and dP/dtmax
The calculated dPa/dtmax and dPlv/dtmax are plotted against heart rate, aortic pulse pressure
and left ventricle pulse pressure in figure 4.5 below. The measurements are taken in baseline
for the following 7 patients: loc6, loc7, loc 8, loc11, loc12, loc21 and loc27.
14
Figure 4.5 The maximum rate of pressure rise in the aorta and left ventricle plotted against left ventricular
pulse pressure (PPlv), aortic pulse pressure (PPa) and heart rate (HR) for seven patients in baseline.
When looking at the heart rate dependency of the maxima in rate of pressure rise in the
lower graphs of figure 4.5 the conclusion made earlier still looks to be true. There seems to
be no relation between the heart rate and dPa/dtmax or dPlv/dtmax.
Looking at the heart rates in rest patient loc27 has a heart rate of approximately 53 bpm
differing from the other patients who have heart rates of around 68 bpm. This is an
unusually low heart rate but could be because of the patient being an athlete or having
bradycardia.
The upper graphs in figure 4.5 show that there could be a linear relationship between the
maxima in pressure rise and their corresponding pulse pressures.
For the same patients these same properties are calculated and displayed in figure 4.6 below
with the condition that the aortic pressure is measured in a hyperemic artery.
15
Figure 4.6 The maximum rate of pressure rise in the aorta and left ventricle plotted against left ventricular
pulse pressure (PPlv), aortic pulse pressure (PPa) and heart rate (HR) for seven patients in rest where the aortic
pressure is measured in a hyperemic artery.
Comparing the heart rate graphs in figure 4.6 with the same graphs in figure 4.5 the heart
rate seems to have become more irregular in all patients. But the difference in heart rate is
not large enough to be significant and a slightly irregular heart beat is normal for human
beings.
Looking at the pulse pressure graphs in the same figures the only difference one notices is
that dPa/dtmax is significantly higher in hyperemia in comparison to baseline for the patient
loc27. But when comparing the other patients some have lowered and some have higher
dPa/dtmax in hyperemia compared to baseline. So there is no consistent difference and thus
would imply that the maxima in rate of pressure rise are not affected by the increased blood
flow in the coronary artery.
Finally for the same patients these same properties are calculated while the patients are
performing the Valsalva maneuver. This data is displayed in figure 4.7, 4.8 and 4.9 below.
16
Figure 4.7 The maximum rate of pressure rise in the aorta and left ventricle plotted against left ventricular
pulse pressure (PPlv), aortic pulse pressure (PPa) and heart rate (HR) for patients loc6 and loc7 while the
patients do the Valsalva Maneuver. Data is separated in two parts where strain (VMS) is when the patient
exerts a pressure on the chest and no strain (VMNS) where the patient relaxes.
Figure 4.8 The maximum rate of pressure rise in the aorta and left ventricle plotted against left ventricular
pulse pressure (PPlv), aortic pulse pressure (PPa) and heart rate (HR) for patients loc8 and loc12 while the
patients do the Valsalva Maneuver. Data is separated in two parts where strain (VMS) is when the patient
exerts a pressure on the chest and no strain (VMNS) where the patient relaxes.
17
Figure 4.9 The maximum rate of pressure rise in the aorta and left ventricle plotted against left ventricular
pulse pressure (PPlv), aortic pulse pressure (PPa) and heart rate (HR) for patients loc21 and loc27 while the
patients do the Valsalva Maneuver. Data is separated in two parts where strain (VMS) is when the patient
exerts a pressure on the chest and no strain (VMNS) where the patient relaxes.
The main difference between the data with strain and the data without strain seems to be
that the strain data has a higher variance in pulse pressure. This is to be expected because by
doing the Valsalva maneuver one alters the output of the heart and thus alters the pulse
pressure.
If one looks at the overall data including strain and no strain a different conclusion is found
as in the other two conditions: heart rate seems to be inversely proportional to dPa/dtmax
and dPlv/dtmax. Furthermore a probable linear relationship between the maxima in pressure
rise and their corresponding pulse pressures can again be concluded.
In conclusion aortic pulse pressure is shown to have a probable linear relationship with
dPa/dtmax. Because of this property aortic pulse pressure is used as a coefficient in a linear
regression model to find a significant relationship between dPa/dtmax and dPlv/dtmax.
Heart rate has little effect on dPa/dtmax or dPlv/dtmax in the conditions of baseline and
hyperemia and is thus not used as a coefficient in these conditions. In the Valsalva maneuver
condition the heart rate is used as a coefficient because of the probable linear relationship
between the two.
4.4 Relationship between aortic pulse pressure and dPa/dtmax/dPlv/dtmax
ratio
To look for the relationship between the ratio dPa/dtmax/dPlv/dtmax and aortic pulse pressure
linear regression is used in SPSS. The outcome of the linear regression for the data in the
three conditions is displayed below in tables 4.10, 4.11 and 4.12.
18
Linear regression baseline
Constant
Aortic pulse pressure
B
0,328
0,002*
SE B
0,061
0,001
Note R = 0,233, R2 = 0,054, * p = 0,066.
Table 4.10 The outcome of multiple linear regression when relating the ratio dPa/dtmax/dPlv/dtmax through the
aortic pulse pressure for 7 patients in baseline.
Linear regression hyperemia
Constant
Aortic pulse pressure
B
0,261
0,005*
SE B
0,084
0,001
Note R = 0,417, R2 = 0,174, * p = 0,001.
Table 4.11 The outcome of multiple linear regression when relating the ratio dPa/dtmax/dPlv/dtmax through the
aortic pulse pressure and left ventricle pulse pressure for 7 patients where the aortic pressure is measured in a
hyperemic artery.
Multiple linear regression Valsalva maneuver
Constant
Aortic pulse pressure
Heart rate
B
1,453
0,000*
-0,015**
SE B
0,241
0,001
0,003
Note R = 0,473, R2 = 0,224, * p = 0,940, ** p < 0,001.
Table 4.12 The outcome of multiple regression when relating the ratio dPa/dtmax/dPlv/dtmax through the aortic
pulse pressure and left ventricle pulse pressure for 7 patients who are performing the Valsalva maneuver
during strain.
In the above tables the B stands for the coefficient values, so in the baseline regression the
value of β0 is 0,328 and the value of β1 is 0,002 in our model equation [5]. SE B stands for the
standard error in the coefficient B. R is the correlation between the predicted and the
observed values. So the R gives an indication of the strength of the relationship. R ranges
from 0.1 to 1.0 and in general8 is considered small for 0.1 to 0.29, medium for 0.30 to 0.49
and large for 0.5 to 1.0. R2 is the coefficient of determination and gives an indication how
much variance the two variables share, in this case aortic pulse pressure and the ratio
dPa/dtmax/dPlv/dtmax. p Is called the significance level and this indicates how much
confidence one should have in the found relationship. So a value of 0.01 for p means that
there is only a one percent chance that there really is no relationship. For p values up to 0.05
the relationship is significant, for p = 0.05 till 0.1 the relationship is considered to be
marginal and above 0.1 considered to be non significant.
Looking at the linear regression models of the hyperemia and baseline conditions one first
notices that the coefficient values and their standard errors like quite similar. This is good
because ideally one would want the same model for every situation. The multiple linear
19
regression model in the Valsalva condition shows very different coefficients because of the
inclusion of heart rate as a coefficient. The aortic pulse pressure coefficient is 0 and has a
very high p indicating that heart rate is the sole contributor to the relationship.
The problem with the model in the condition of baseline is that it is not quite significant
enough (p = 0,066), the strength of the relationship is pretty low (R = 0,233) and the shared
variance is very low (R2 = 0,054). The linear regression models in hyperemia and Valsalva fit
the data better with a higher R, R2 and p < 0,05 but R and R2 are still quite low.
To check the previous assumption that heart rate does not influence the relationship
between dPlv/dtmax and dPa/dtmax multiple linear regression is used in SPSS with aortic pulse
pressure and heart rate as coefficients in the baseline and hyperemia conditions. The results
are shown below in table 4.13 and 4.14.
Multiple linear regression baseline
Constant
Aortic pulse pressure
Heart rate
B
0,020
0,002*
0,004**
SE B
0,259
0,001
0,004
Note R = 0,278, R2 = 0,077, * p = 0,046, ** p = 0,227.
Table 4.13 The outcome of multiple linear regression when relating the ratio dPa/dtmax/dPlv/dtmax through the
aortic pulse pressure for 7 patients in baseline.
Multiple linear regression hyperemia
Constant
Aortic pulse pressure
Heart rate
B
1,226
0,003*
-0,013**
SE B
0,693
0,002
0,009
Note R = 0,473, R2 = 0,224, * p = 0,031, ** p < 0,166.
Table 4.14 The outcome of multiple linear regression when relating the ratio dPa/dtmax/dPlv/dtmax through the
aortic pulse pressure and left ventricle pulse pressure for 7 patients where the aortic pressure is measured in a
hyperemic artery.
Looking at the above tables one notices that the R and R2 have both increased a little but at
the cost of increasing the p and thus decreasing the significance of the model. So the
inclusion of heart rate in the models is no real advantage in de hyperemia and baseline
conditions.
To see if these linear models can be used in practice dPlv/dtmax is calculated with the ratio
dPa/dtmax/dPlv/dtmax found by the linear model’s equation and comparing this with the
measured dPlv/dtmax.
For example in the case of baseline the following equation is used:
20
In this equation the aortic pulse pressure is inserted to calculate the corresponding ratio.
The measured dPlv/dtmax is divided by this ratio and thus approximating the dPlv/dtmax. In
figures 4.15, 4.16 and 4.17 below this estimated dPlv/dtmax is plotted against the measured
dPlv/dtmax. Included in the graphs is the linear relationship between the measured and
estimated dPlv/dtmax as approximated with linear regression.
Figure 4.15 The estimated dPlv/dtmax plotted against the measured dPlv/dtmax for 7 patients in baseline. The
aortic pulse pressure was used to estimate dPlv/dtmax. Properties of the linear regression line: R = 0.22, R2 =
0.05, p = 0.08 and SEE = 198.
21
Figure 4.16 The estimated dPlv/dtmax plotted against the measured dPlv/dtmax for 7 patients where the aortic
pressure is measured in a hyperemic artery. The aortic pulse pressure was used to estimate dPlv/dtmax.
Properties of the linear regression line: R = 0.27, R2 = 0.07, P < 0.05 and SEE = 292.
Figure 4.17 The estimated dPlv/dtmax plotted against the measured dPlv/dtmax for 7 patients who are doing the
Valsalva Maneuver during strain. The aortic pulse pressure was used to estimate dPlv/dtmax. Properties of the
linear regression line: R = 0.08, R2 = 0.00, P = 0.48 and SEE = 351.
22
If the linear regression model used is perfect the data points in the above figures should be
on the line x = y. None of the models come real close to giving a good approximation. In the
Valsalva maneuver condition the R is too low to even indicate a relationship. In hyperemia
and in baseline the significance is higher and the R is greater and thus the correlation is
bigger but it is still just a small correlation.
Finally one can display the variation in the data by using Bland-Altman plots. Below in figures
4.18, 4.19 and 4.20 the Bland-Altman plots are displayed for the data used in this research.
Figure 4.18 The difference between the measured and estimated dPlv/dtmax is plotted against the mean of the
measured and estimated dPlv/dtmax for 7 patients in baseline.
23
Figure 4.19 The difference between the measured and estimated dPlv/dtmax is plotted against the mean of the
measured and estimated dPlv/dtmax for 7 patients where the aortic pressure is measured in a hyperemic artery.
Figure 4.20 The difference between the measured and estimated ratios is plotted against the mean of the
measured and estimated ratios for 7 patients who are doing the Valsalva Maneuver during strain.
Looking at the above figures one notices that in the baseline and hyperemia conditions that
at lower means the measured dPlv/dtmax is underestimated and at higher means the
24
measured dPlv/dtmax is overestimated by the linear regression. In the Valsalva maneuver plot
no real conclusion can be drawn. For all three figures the difference between estimated and
measured is also quite large for all mean values indicating no real good relationships.
5
Discussion
5.1
Coefficient of heart rate in multiple linear regression
In the results the effect of heart rate in baseline and hyperemia on dPa/dtmax/dPlv/dtmax was
seen to be negligible but in the Valsalva maneuver the heart rate was the only significant
contributor. In normal conditions there should be no to little influence of the heart rate on
the contractility. The reason why it is significant when the patient is doing the Valsalva
maneuver is because during the Valsalva maneuver the heart rate is influenced by the aortic
pulse pressure due to the baroreceptor reflex. Heart rate decreases when the aortic
pressure is higher and increases when the aortic pressure lowers. So the heart rate and
aortic pulse pressure both represent the same dependency and so one of the two can be
used in a single regressional analyses.
5.2
Data analysis issues
The most important issue in the data analysis is the time interval in the given pressure
measurements. Due to the time interval between pressure measurements being too long
(0,005 s) the maximum rate of pressure rise in the aorta in a few patients in the condition of
hyperemia had to be discarded. Because of the very large rise in pressure in these patients
during the main peak and the low sampling rate the maximum rate of pressure rise is greatly
overestimated by the derivative approximation.
5.3
Overall relationship between measured and estimated dPlv/dtmax
In this research a consistently moderate relationship is found between the measured and
estimated dPlv/dtmax in both the baseline and hyperemia conditions taking into account the
aortic pulse pressure.
When the patients do the Valsalva maneuver a different relationship is found using the heart
rate but with a significantly lower accuracy. This could be because by doing the Valsalva
maneuver one alters properties of the heart like the stroke volume. These changes cannot all
be accounted for by the pulse pressures. Also the other properties of the arteries like the
characteristic impedance could have a bigger influence on the relationship in this condition
compared to hyperemia and baseline.
This study was done with a limited amount of patients and the relationship between the
estimated and measured dPlv/dtmax was determined with only the aortic pulse pressure and
heart rate. The research was done on diseased patients but the aortic pressure was
determined in healthy coronary arteries. In future studies the relationship should be
examined in a broader scope with patients who have diseases with varying degrees of
sickness and more total patients. Secondly the study should incorporate a bigger range of
vascular properties including the characteristic impedance.
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As was seen in the literary review a similar study was done in 2009 but their relationship
between the estimated and measured dPlv/dtmax was significantly better with R = 0,89, SEE =
330 and p < 0,0001. Comparing that to the linear models in this research which are
moderate at best it would suggest that the mean arterial pressure and characteristic
impedance are better determinants to use then aortic pulse pressure and heart rate to
define an accurate enough relationship between dPa/dtmax and dPlv/dtmax.
Combining these results the conclusion is that there is a possibility that the contractility of
the heart can be determined by using the dPa/dtmax in various conditions with patients
having various diseases. However the calculations of the vascular properties add some
complexity and work.
7
Conclusion
The maximum rate of pressure rise in the aorta does not relate significantly enough to the
maximum rate of pressure rise in the left ventricle through the aortic pulse pressure and/or
the heart rate to be used in clinical application. This relationship cannot be used to
determine the contractility of the heart accurately enough with only the pulse pressure and
heart rate. This conclusion is the same in all conditions.
8
References
1. Quinones MA, Gaasch WH, Alexander JK. Influence of acute changes in preload,
afterload, contractile state and heart rate on ejection and isovolumic indices of
myocardial contractility in man. Circulation 1976; 53: 293-302.
2. Liu Z, Brin KP, Yin FC. Estimation of total arterial compliance: An improved method
and evaluation of current methods. Am J Physiol 1986; 251: H588-H600.
3. Taylor SH, Snow HM, Linden RJ. Relationship between left ventricular
and aortic dP-dt (max). Proc R Soc Med 1972; 65: 550 – 552.
4. Bargiggia GS, Bertucci C, Recusani F, Raisaro A, de Servi S, Valdes- Cruz LM, et al. A
new method for estimating left ventricular dP/dt by continuous wave Dopplerechocardiography: Validation studies at cardiac catheterization. Circulation 1989; 80:
1287 – 1292.
5. Chen C, Rodriguez L, Guerrero JL, Marshall S, Levine RA, Weyman AE, et al.
Noninvasive estimation of the instantaneous first derivative of left ventricular
pressure using continuous-wave Doppler echocardiography. Circulation 1991; 83:
2101 – 2110.
6. Chung N, Nishimura RA, Holmes DR Jr, Tajik AJ. Measurement of left ventricular dp/dt
by simultaneous Doppler echocardiography and cardiac catheterization. J Am Soc
Echocardiogr 1992; 5: 147 – 152.
7. Masutani S, Iwamoto Y, Ishido H, Senzaki H. Relationship of Maximum Rate of
Pressure Rise Between Aorta and Left Ventricle in Pediatric Patients. Circ J 2009; 73:
1698-1704.
8. Cohen J. Statistical Power Analysis for the Behavioral Sciences; 1988: 79-81.
26
A
Taylor series method
To calculate the central first order derivate using a fourth order polynomial one can use the
Taylor series method. In this method one first looks at 5 specified base points including the
to be evaluated base point xi, two base points before this xi-1, xi-2 and two after xi+1, xi+2. In
the formulas below h is the fixed interval. The derivative can be found by expanding the
Taylor series around these points as follows:
This can be expressed in matrix form like so:
To solve the derivates one multiplies both sides of the equation with the inverse of the
coefficient matrix to the left. This gives:
Finally the central first order derivative becomes:
27
The forward and backward derivatives are derived by the same method but with only taking
base points after and previous the base point one is evaluating.
B
Delphi program code
unit SDIMAIN;
interface
uses Windows, Classes, Graphics, Forms, Controls, Menus,
Dialogs, StdCtrls, Buttons, ExtCtrls, ComCtrls, ImgList, StdActns,
ActnList, ToolWin, SysUtils;
type
TSDIAppForm = class(TForm)
OpenDialog: TOpenDialog;
SaveDialog: TSaveDialog;
ToolBar1: TToolBar;
ToolButton9: TToolButton;
ToolButton1: TToolButton;
ToolButton2: TToolButton;
ToolButton3: TToolButton;
ToolButton4: TToolButton;
ToolButton5: TToolButton;
ToolButton6: TToolButton;
ActionList1: TActionList;
FileNew1: TAction;
FileOpen1: TAction;
FileSave1: TAction;
FileSaveAs1: TAction;
FileExit1: TAction;
EditCut1: TEditCut;
EditCopy1: TEditCopy;
EditPaste1: TEditPaste;
HelpAbout1: TAction;
StatusBar: TStatusBar;
ImageList1: TImageList;
MainMenu1: TMainMenu;
File1: TMenuItem;
FileNewItem: TMenuItem;
FileOpenItem: TMenuItem;
FileSaveItem: TMenuItem;
FileSaveAsItem: TMenuItem;
N1: TMenuItem;
FileExitItem: TMenuItem;
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Edit1: TMenuItem;
CutItem: TMenuItem;
CopyItem: TMenuItem;
PasteItem: TMenuItem;
Help1: TMenuItem;
HelpAboutItem: TMenuItem;
Button1: TButton;
procedure FileNew1Execute(Sender: TObject);
procedure FileOpen1Execute(Sender: TObject);
procedure FileSave1Execute(Sender: TObject);
procedure FileExit1Execute(Sender: TObject);
procedure HelpAbout1Execute(Sender: TObject);
procedure FormCreate(Sender: TObject);
procedure Button1Click(Sender: TObject);
private
{ Private declarations }
public
{ Public declarations }
end;
var
SDIAppForm: TSDIAppForm;
F:TStrings ;
implementation
uses about;
{$R *.dfm}
procedure TSDIAppForm.FileNew1Execute(Sender: TObject);
begin
{ Do nothing }
end;
procedure TSDIAppForm.FileOpen1Execute(Sender: TObject);
begin
OpenDialog.Execute;
end;
procedure TSDIAppForm.FileSave1Execute(Sender: TObject);
begin
SaveDialog.Execute;
end;
procedure TSDIAppForm.Button1Click(Sender: TObject);
var
dpdcsl,dpacsl,dpadtsl,dpvcsl,Pasortsl,Plvsortsl,sl2,cycle,graphtimesl,graphsl,sort2,sort3,sort4,timesl:
TStringList;
i,j,k,l,m,n,o,p,q: integer;
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Dir,max,max2,max3,max4,amin,mini,dpadt,hr,dpaa,padiff,ratio,Plvsort,Plvmax,Plvmin,Plvdiff,Pasort,
Pamax,Pamin,dpva,dpdc,dpac,dpvc,time,maxdiff1,maxdiff2,methdiffm,methdifft: string;
openDialog : TOpenDialog; // Open dialog variable
begin
begin
// Create the open dialog object - assign to our open dialog variable
openDialog := TOpenDialog.Create(self);
// Set up the starting directory to be the current one
openDialog.InitialDir := GetCurrentDir;
// Only allow existing files to be selected
openDialog.Options := [ofFileMustExist];
// Display the open file dialog
if openDialog.Execute
then ShowMessage('File : '+openDialog.FileName)
else ShowMessage('Open file was cancelled');
end;
sl2 := TstringList.Create;
dPdcsl := TstringList.Create;
graphtimesl := TstringList.Create;
Pasortsl := TstringList.Create;
Plvsortsl := TstringList.Create;
graphsl := TstringList.Create;
graphsl.Append('Time PLV' + #9 +'Time PA' + #9 + 'Time Pd' + #9 +'dPLV/dt max'+#9+'dPa/dt max'
+#9+'dPd/dt max');
graphtimesl.Append('dPa/dt max/dPLV/dt max'+#9+ 'Aortic pressure difference' +#9+ 'Left ventricle
pulse pressure'+#9+'Heart rate');
cycle := TstringList.Create;
dpacsl := TstringList.Create;
dpvcsl := TstringList.Create;
sort2 := TstringList.Create;
sort3 := TstringList.Create;
sort4 := TstringList.Create;
timesl := TstringList.Create;
sl2.LoadFromFile(openDialog.Filename);
sl2.Text := StringReplace(sl2.Text,#9,Char(13)+Char(10),[rfReplaceAll]);
for j:= sl2.Count-1 downto 0 do
begin
if Trim(sl2[j]) = '' then
sl2.Delete(j);
end;
for k := 0 to sl2.Count - 1 do
if sl2[k] = '1' then
begin
cycle.Add(IntToStr(k));
end;
for k := 0 to cycle.Count-2 do
30
begin
for m := 1 to Trunc((StrToInt(cycle[k+1])-StrToInt(cycle[k]))/6) do
begin
Pasort := sl2[StrToInt(cycle[k])+2+6*m];
Pasortsl.Add(Pasort);
end;
amin := '-1000000000000000000000000';
for n := 0 to Pasortsl.Count - 1 do
begin
if StrToFloat(Pasortsl[n]) > StrToFloat(amin) then
amin := Pasortsl[n];
end;
mini := IntToStr(Pasortsl.IndexOf(amin));
Pasortsl.Clear;
for i := 10 to StrToInt(mini) do
begin
dPac := FloatToStr((1/0.005)*((1/12)*StrToFloat(sl2[StrToInt(cycle[k])-10+6*i]) (2/3)*StrToFloat(sl2[StrToInt(cycle[k])-4+6*i])+(2/3)*StrToFloat(sl2[StrToInt(cycle[k])+8+6*i])(1/12)*StrToFloat(sl2[StrToInt(cycle[k])+14+6*i])));
dPdc := FloatToStr((1/0.005)*((1/12)*StrToFloat(sl2[StrToInt(cycle[k])-9+6*i]) (2/3)*StrToFloat(sl2[StrToInt(cycle[k])-3+6*i])+(2/3)*StrToFloat(sl2[StrToInt(cycle[k])+9+6*i])(1/12)*StrToFloat(sl2[StrToInt(cycle[k])+15+6*i])));
dPvc := FloatToStr((1/0.005)*((1/12)*StrToFloat(sl2[StrToInt(cycle[k])-6+6*i]) (2/3)*StrToFloat(sl2[StrToInt(cycle[k])+6*i])+(2/3)*StrToFloat(sl2[StrToInt(cycle[k])+12+6*i])(1/12)*StrToFloat(sl2[StrToInt(cycle[k])+18+6*i])));
Pasort := sl2[StrToInt(cycle[k])+2+6*i];
Plvsort := sl2[StrToInt(cycle[k])+6+6*i];
time := sl2[StrToInt(cycle[k])+1+6*i];
dPacsl.Add(dPac);
dPdcsl.Add(dPdc);
dPvcsl.Add(dPvc);
Pasortsl.Add(Pasort);
Plvsortsl.Add(Plvsort);
Sort2.Add(dPdc);
Sort3.Add(dPac);
Sort4.Add(dPvc);
timesl.Add(time);
end;
Pamax:= '-10000000';
for p := 0 to Pasortsl.Count - 1 do
begin
if StrToFloat(Pasortsl[p]) > StrToFloat(Pamax) then
Pamax := Pasortsl[p];
end;
Plvmax:= '-10000000';
for p := 0 to Pasortsl.Count - 1 do
begin
31
if StrToFloat(Plvsortsl[p]) > StrToFloat(Plvmax) then
Plvmax := Plvsortsl[p];
end;
Pamin := '1000000000000000000000000';
for j := 0 to Pasortsl.Count - 1 do
begin
if StrToFloat(Pasortsl[j]) < StrToFloat(Pamin) then
Pamin := Pasortsl[j];
end;
Plvmin := '1000000000000000000000000';
for j := 0 to Pasortsl.Count - 1 do
begin
if StrToFloat(Plvsortsl[j]) < StrToFloat(Plvmin) then
Plvmin := Plvsortsl[j];
end;
max2 := '-10000000';
for q := 0 to Sort2.Count - 1 do
begin
if StrToFloat(Sort2[q]) > StrToFloat(max2) then
max2 := Sort2[q];
end;
max3 := '-10000000';
for l := 0 to Sort3.Count - 1 do
begin
if StrToFloat(Sort3[l]) > StrToFloat(max3) then
max3 := Sort3[l];
end;
max4 := '-10000000';
for l := 0 to Sort4.Count - 1 do
begin
if StrToFloat(Sort4[l]) > StrToFloat(max4) then
max4 := Sort4[l];
end;
Padiff := FloatToStr(StrToFloat(Pamax)-StrToFloat(Pamin));
Plvdiff := FloatToStr(StrToFloat(Plvmax)-StrToFloat(Plvmin));
max2i := timesl[dPdcsl.IndexOf(max2)];
max3i := timesl[dPacsl.IndexOf(max3)];
max4i := timesl[dPvcsl.IndexOf(max4)];
ratio := FloatToStr(StrToFloat(max3)/StrToFloat(max4));
hr := FloatToStr(60/(StrToFloat(sl2[StrToInt(cycle[k+1])+1])-StrToFloat(sl2[StrToInt(cycle[k])+1])));
graphsl.Add(max4i+#9+max3i+#9+max2i+#9+max4+#9+max3+#9+max2);
graphtimesl.Add(ratio+#9+Padiff+#9+Plvdiff+#9+hr);
dPacsl.Clear;
dPvcsl.Clear;
dPdcsl.Clear;
Pasortsl.Clear;
32
Plvsortsl.Clear;
Sort2.Clear;
Sort3.Clear;
Sort4.Clear;
timesl.Clear;
end;
graphsl.SaveToFile(openDialog.FileName+'.dpalvgraph.txt');
graphtimesl.SaveToFile(openDialog.FileName+'.timediffgraph.txt');
end;
procedure TSDIAppform.FormCreate(Sender: TObject);
begin
{ Do nothing }
end;
procedure TSDIAppForm.FileExit1Execute(Sender: TObject);
begin
Close;
end;
procedure TSDIAppForm.HelpAbout1Execute(Sender: TObject);
begin
AboutBox.ShowModal;
end;
end.
33